If mail domain is hosted locally, Postfix will verify whether sender/recipient under same domain exists locally. If not, it rejects the mail with error like 'Sender address rejected: User unknown in virtual mailbox table'.
*) How about create mail accounts on new server to avoid this error?
*) Or, alternatively, in /etc/postfix/main.cf, comment out 'smtpd_reject_unlisted_recipient=yes' and 'smtpd_reject_unlisted_sender=yes', remove 'reject_unlisted_recipient' and 'reject_unlisted_sender'. then restart Postfix. This is NOT recommended, and you'd better enable them after testing.